The General Circulation of the Atmosphere Tapio Schneider

Overview Aims Axisymmetric features of Earth’s atmosphere Tropical Hadley Circulation –Hide’s theorem Extratropical Circulation Atmospheric Macroturbulence

Aims Require a theory of general circulation of the atmosphere to produce models of the Earth’s atmosphere, both past, future and for atmospheric models of other planets. A general circulation theory for idealised atmospheres with axisymmetric rotations is a prerequisite for any future more complete, general circulation theory, which must be reducible to this canonical case. To draw attention to unresolved, fundamental questions about the general circulation of dry atmospheres, questions whose resolution is a prerequisite for any general circulation theory, moist or dry.

Axisymmetric Circulation

Temporal and Zonal Circulations

Axisymmetric Flow Proposed by Hadley Axisymmetric circulation  baroclinically unstable Eddies transport heat polewards

Macroturbulence Mactroturbulence – large scale eddies, +1000km. Eddies produced by baroclinic instability. Transport angular momentum into latitude zones in which they are created. Angular momentum flux into zone compensated by surface drag  surface westerlies appear in baroclinic zones into which angular momentum is being transported. Vertical structure of winds and strength of upper level jets linked to surface winds by thermal/gradient wind balance.

Thermal Wind Relates vertical shear of the zonal wind to meridional temperature. Not actually a wind, but the difference in the geostrophic wind between two pressure levels p1 and p0, with p1 < p0. Only present in an atmosphere with horizontal gradients of temperature i.e. baroclinic. Flows around areas of high and low temperature as the geostrophic wind flows around areas of high and low pressure.

Axisymmetric Circulation Vs. Macroturbulence

Explanation of Figure 3 Bottom row fig 3  temporal and zonal means of mass flux stream function and angular momentum in steady states of macroturbulent circulation that correspond to the axisymmetric circulation in top row. Macroturbulent Hadley cells extend further poleward than axisymmetric simulations. Streamlines in upper parts of Hadley cell cut angular momentum contours. Local Rossby numbers reduced relative to axisymmetric circulation. Eddies strengthen the equinoctial Hadley cells (3a and 3b) and weaken the winter cell (3b and 3e) Mass flux in Hadley cells in macroturbulent model same order of magnitude as in Earth’s Hadley cells. When max heating moved to 6 degrees latitude, winter cell 1.5 times bigger and summer cell 1.5 times smaller (3d and e).

Implications of Hide’s Theorem u <= u m = Ωa sin 2 (  )/cos(  ) Assume gradient-wind balance, then from meridional momentum equation:   Ф <= 2 Ω 2 a 2  3 Ф=gz, (assuming small latitude = tropics) Use ideal-gas result p = p 0 exp(-Ф/RT)(T is vertically averaged) => constraints on meridional decrease in temperature Assume T ~  h cos 2  h = pole-equator T difference Then Hadley circulation extends to  m ~ sqrt (gz *  h ) / (Ω 2 a 2 T 0 )

Meridional Extent of Hadley Cells

Entropy – measures amount of disorder in a system For an ideal gas: s = c p ln (T p –R/c )   =  0 exp(-s/c p )   constant  s constant Potential vorticity – measure of vorticity, normalized by entropy P = (planetary vorticity + relative vorticity) / (width of entropy contour) = (f +  )/H Conserved quantity for adiabatic processes Potential Vorticity & Entropy

Isentropic Mass Circulation Eulerian mass flux  Isentropic mass flux  Extratropical flow ~ large-scale eddies ~ adiabatic  convenient to use isentropic coordinates Entropy transported poleward Eddy entropy flux >> mean entropy flux ii bb Isentropic, meridional mass flux Isentropic eddy flux of potential vorticity P Eddy flux of  at sfc (boundary term) Ekman mass flux Assume eddies mix P downgradient &   P>0 in interior  southward P flux  …

Assume eddies mix potential vorticity & potential temp. diffusively Assume there is  e so that above  e, atmos. is in radiative- convective equilibrium. Integrate previous eqn.  LHS vanishes, ignore Ekman flux  find up to which entropy fluxes are significant – this level must be lower than the tropopause  p e >= p t  Turbulence as a diffusive process  1 bulk stability supercriticality – measure of vertical extent of eddy entropy fluxes

Supercriticality constraint x-axis – negative  gradient ~ entropy gradient y-axis – bulk stability S c <1 regime – eddy entropy fluxes weak, tropopause set by radiation/convection S c ~1 regime – eddy entropy fluxes large & stabilize the thermal stratification  tropopause height adjusted  A state with strong nonlinear eddy-eddy interactions (S c >>1) adjusts thermal stratification so that S c <~1 (and has weak eddy- eddy interactions)

Summary Differential heating causes Hadley circulation in tropics, Polar cell near poles In midlatitudes, differential heating causes baroclinic instability Hide’s Theorem imposes upper limit to Hadley circulation extent Extratropical circulation associated with (adiabatic) eddy fluxes of P,  If eddies act diffusively, supercriticality <=1 –thermal stratification / tropopause height linked to eddy strength